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## zero-one polynomial

Help
momo
2012-02-13
2012-11-20

• momo
2012-02-13

Hello,
I have a problem: all of a polynomial real zero-finding.
The problem is as follows:
Be a Real polynomial p (x) = (am) * x ** m + (am-1) * x ** m-1 + …. + a1 * x + a0
with m = 2 * k and real parameters (Am. .. a0)
are all zeros of p (x) all real?
i.e. the output to be rlqe (by quantifier elimination) must be true or false!
I have tried to solve with realroots and r_solve but no success.
Thank you very much.

• Thomas Sturm
2012-02-13

I do not quite understand you problem.

If you have got concrete coefficients, then you can use "realroots" and compare the number of roots found with the degree.

If you consider a generic problem with all the a_i being parameters, then you can for fixed degree m use real quantifier elimination on

```ex({x1,...,xm}, x1<>x2 and x1<>x3 and ... and x_m-1<>x_m and p(x_1)=0 and ... and p(x_m)=0)
```

,

but for absolutely generic problems you will not get very far.

Thomas

• momo
2012-02-13

my idea is:
If I have a polynomial of degree m and I m getting zero the real.
Then the problem is solved.
Only here is the problem that I work with general real parammetern

• Thomas Sturm
2012-02-13

Quantifier elimination would give you necessary and sufficient conditions in the parameters. Can you post concrete input, and carefully formulate the question that you have on that input?

Thomas

• momo
2012-02-13

Hallo Thomas,
Input : something real polynomial with parameters;
Output : all the real zero the polynomial
Thanks